A total curvature estimate of closed hypersurfaces in non-positively curved symmetric spaces
Jiangtao Li, Zuo Lin, Liang Xu
TL;DR
The paper tackles the problem of deriving a total curvature estimate for closed hypersurfaces in simply-connected non-positively curved symmetric spaces, linking this estimate to isoperimetric-type inequalities. It develops a Busemann-function framework and constructs a generalized Gauss map with Lipschitz regularity, then bounds the Jacobian of this map by the Gauss-Kronecker curvature, using the area formula to deduce a global lower bound. The main result is the inequality $\displaystyle \int_M |GK| \, d\mathrm{area}_M \ge e^{-n(n+1)\kappa D} \mathrm{area}(\mathbb{S}^n)$, from which a Willmore-type bound and a diameter-controlled isoperimetric inequality in dimension $n+1\le 7$ follow; these yield quantitative isoperimetric control in broad non-positively curved symmetric spaces. Overall, the work extends total-curvature methods to non-positively curved symmetric spaces and provides explicit exponential-in-$\kappa D$ bounds that enhance our understanding of geometric inequalities in such ambient geometries.
Abstract
In this paper, we prove a total curvature estimate of closed hypersurfaces in simply-connected non-positively curved symmetric spaces, and as a corollary, we obtain an isoperimetric inequality for such manifolds.